The scaling study shows that the \(2\omega _{pe}\) -radiation peak is pervasive over broad ranges of \(\alpha \)  and \(n_i\)  [see Fig. REF (c)]. The optimal positron fraction is \(\alpha \sim 2\%\)  at which the \(2\omega _{pe}\) -radiation power is maximized. The maximum \(2\omega _{pe}\) -radiation enhancement exceeds 5 orders of magnitude as shown in Fig. REF (a). The beam-plasma instability generally depends on the beam density \(n_p\)  and its relative velocity spread \(\Delta v_p/v_p\)  [1]}, [2]}, because the linear growth time for the wave excitation is \(t_g\propto n_p^{-1}(\Delta v_p/v_p)^2n_e^{1/2}\)  and the bandwidth of the excited wave scales as \(\Delta \omega /\omega _{0}\sim (\Delta v_p/v_p)(n_e/n_c)^{1/2}\) . In the low \(\alpha \)  regime, the positron beam is formed with a low temperature/emittance, i.e., small dispersion in both coordinate and velocity spaces, producing narrow \(n\omega _{pe}\)  spectral peaks. The \(2\omega _{pe}\) -radiation power in this regime increases with an approximate power-law dependence on \(\alpha \)  [see Fig. REF (c) for \(\alpha \lesssim 2\%\) ]. The lowest positron density, which can result in a notable \(2\omega _{pe}\) -peak, is \(n_p\sim 0.005n_c\) . In the high \(\alpha \)  regime, the bulk positron velocity decreases with \(\alpha \) . The pile-up of the laser-pushed positrons will lead to the slowdown or even breakdown of the positron beam, significantly increasing its temperature/emittance. The bandwidth of the excited plasma waves is broadened, resulting in the drop and eventual saturation of the \(2\omega _{pe}\) -radiation power.
